How well do third-order aperture mass statistics separate E- and B-modes?

With 3rd-order statistics of gravitational shear it will be possible to extract valuable cosmological information from ongoing and future weak lensing surveys which is not contained in standard 2nd-order statistics, due to the non-Gaussianity of the shear field. Aperture mass statistics are an appropriate choice for 3rd-order statistics due to their simple form and their ability to separate E- and B-modes of the shear. However, it has been demonstrated that 2nd-order aperture mass statistics suffer from E-/B-mode mixing because it is impossible to reliably estimate the shapes of close pairs of galaxies. This finding has triggered developments of several new 2nd-order statistical measures for cosmic shear. Whether the same developments are needed for 3rd-order shear statistics is largely determined by how severe this E-/B-mixing is for 3rd-order statistics. We test 3rd-order aperture mass statistics against E-/B-mode mixing, and find that the level of contamination is well-described by a function of $\theta/\theta_{\rm min}$, with $\theta_{\rm min}$ being the cut-off scale. At angular scales of $\theta > 10 \;\theta_{\rm min}$, the decrease in the E-mode signal due to E-/B-mode mixing is smaller than 1 percent, and the leakage into B-modes is even less. For typical small-scale cut-offs this E-/B-mixing is negligible on scales larger than a few arcminutes. Therefore, 3rd-order aperture mass statistics can safely be used to separate E- and B-modes and infer cosmological information, for ground-based surveys as well as forthcoming space-based surveys such as Euclid.Comment: references added, A&A publishe

On the Probability Distributions of Ellipticity

In this paper we derive an exact full expression for the 2D probability distribution of the ellipticity of an object measured from data, only assuming Gaussian noise in pixel values. This is a generalisation of the probability distribution for the ratio of single random variables, that is well-known, to the multivariate case. This expression is derived within the context of the measurement of weak gravitational lensing from noisy galaxy images. We find that the third flattening, or epsilon-ellipticity, has a biased maximum likelihood but an unbiased mean; and that the third eccentricity, or normalised polarisation chi, has both a biased maximum likelihood and a biased mean. The very fact that the bias in the ellipticity is itself a function of the ellipticity requires an accurate knowledge of the intrinsic ellipticity distribution of the galaxies in order to properly calibrate shear measurements. We use this expression to explore strategies for calibration of biases caused by measurement processes in weak gravitational lensing. We find that upcoming weak lensing surveys like KiDS or DES require calibration fields of order of several square degrees and 1.2 magnitude deeper than the wide survey in order to correct for the noise bias. Future surveys like Euclid will require calibration fields of order 40 square degree and several magnitude deeper than the wide survey. We also investigate the use of the Stokes parameters to estimate the shear as an alternative to the ellipticity. We find that they can provide unbiased shear estimates at the cost of a very large variance in the measurement. The python code used to compute the distributions presented in the paper and to perform the numerical calculations are available on request.Comment: 24 pages, 18 figures, 2 Tables. Accepted for publication in Monthly Notices of the Royal Astronomical Society Main Journa

Cosmic Shear and the Intrinsic Alignment of Galaxies

Cosmology has recently entered an era of increasingly rich observational data sets, all being in agreement with a cosmological standard model that features only a small number of free parameters. One of the most powerful techniques to constrain these parameters and test the accuracy of the concordance model is the weak gravitational lensing of distant galaxies by the large-scale structure, or cosmic shear. This thesis investigates the optimisation of present and future cosmic shear surveys with respect to the extraction of cosmological information and deals with the characterisation and control of the intrinsic alignment of galaxies, a major systematic in cosmic shear data. A detailed derivation of the covariance of the weak lensing convergence bispectrum is presented, clarifying the relation between existing formalisms, providing illustration, and simplifying the practical computation. The results are then applied to forecasts on cosmological constraints by cosmic shear two- and three-point statistics with the proposed Euclid satellite. Besides, a novel method to assess the impact of unknown systematics on cosmological parameter constraints is summarised, and several aspects concerning the weak lensing analysis of the Hubble Space Telescope COSMOS survey are highlighted. A synopsis of the current state of knowledge about the intrinsic alignment of galaxies is given, including its physical origin, modelling attempts, simulation results, and existing observations. Possible corrections to the prevailing model of intrinsic alignments are suggested, before presenting new observational constraints on matter-intrinsic shear correlations using several galaxy samples from the Sloan Digital Sky Survey. For the first time a data set with only photometric redshift information is included, after developing the formalism for correlation function models that take photometric redshift scatter into account. The intrinsic alignment signal of early-type galaxies is found to increase with galaxy luminosity and to be inconsistent with the default redshift evolution of a widely used model, both with high confidence. Moreover the nulling technique is developed, a method to remove gravitational shear-intrinsic ellipticity correlations from cosmic shear data by solely relying on the well-known redshift dependence of the signals, and its performance on realistically modelled cosmic shear two-point statistics is investigated. Subsequently, the principle of intrinsic alignment boosting, an inverse and likewise geometrical approach capable of extracting the intrinsic alignment signal from cosmic shear data, is derived. Both techniques are shown to robustly remove or isolate the intrinsic alignment signal, but are subject to a significant loss of statistical power caused by the similarity between the redshift dependence of the lensing signal and shear-intrinsic correlations in combination with strict model independence. As an alternative ansatz, the joint analysis of various probes available from cosmic shear surveys is considered, including cosmic shear, galaxy clustering, lensing magnification effects, and cross-correlations between galaxy number densities and shapes. The self-calibration capabilities of intrinsic alignments and the galaxy bias in the combined data are found to be excellent for realistic survey parameters, recovering the constraints on cosmological parameters for a pure cosmic shear signal in presence of flexible parametrisations of intrinsic alignments and galaxy bias with about a hundred nuisance parameters in total

Gaussianisation for fast and accurate inference from cosmological data

We present a method to transform multivariate unimodal non-Gaussian posterior probability densities into approximately Gaussian ones via non-linear mappings, such as Box--Cox transformations and generalisations thereof. This permits an analytical reconstruction of the posterior from a point sample, like a Markov chain, and simplifies the subsequent joint analysis with other experiments. This way, a multivariate posterior density can be reported efficiently, by compressing the information contained in MCMC samples. Further, the model evidence integral (i.e. the marginal likelihood) can be computed analytically. This method is analogous to the search for normal parameters in the cosmic microwave background, but is more general. The search for the optimally Gaussianising transformation is performed computationally through a maximum-likelihood formalism; its quality can be judged by how well the credible regions of the posterior are reproduced. We demonstrate that our method outperforms kernel density estimates in this objective. Further, we select marginal posterior samples from Planck data with several distinct strongly non-Gaussian features, and verify the reproduction of the marginal contours. To demonstrate evidence computation, we Gaussianise the joint distribution of data from weak lensing and baryon acoustic oscillations (BAO), for different cosmological models, and find a preference for flat $\Lambda$CDM. Comparing to values computed with the Savage-Dickey density ratio, and Population Monte Carlo, we find good agreement of our method within the spread of the other two.Comment: 14 pages, 9 figure

Maximal compression of the redshift space galaxy power spectrum and bispectrum

We explore two methods of compressing the redshift space galaxy power spectrum and bispectrum with respect to a chosen set of cosmological parameters. Both methods involve reducing the dimension of the original data-vector ( e.g. 1000 elements ) to the number of cosmological parameters considered ( e.g. seven ) using the Karhunen-Lo\`eve algorithm. In the first case, we run MCMC sampling on the compressed data-vector in order to recover the one-dimensional (1D) and two-dimensional (2D) posterior distributions. The second option, approximately 2000 times faster, works by orthogonalising the parameter space through diagonalisation of the Fisher information matrix before the compression, obtaining the posterior distributions without the need of MCMC sampling. Using these methods for future spectroscopic redshift surveys like DESI, EUCLID and PFS would drastically reduce the number of simulations needed to compute accurate covariance matrices with minimal loss of constraining power. We consider a redshift bin of a DESI-like experiment. Using the power spectrum combined with the bispectrum as a data-vector, both compression methods on average recover the 68% credible regions to within 0.7% and 2% of those resulting from standard MCMC sampling respectively. These confidence intervals are also smaller than the ones obtained using only the power spectrum by (81%, 80%, 82%) respectively for the bias parameter b_1, the growth rate f and the scalar amplitude parameter A_s.Comment: 27 pages, 8 figures, 1 table, Accepted 2018 January 28. Received 2018 January 25; in original form 2017 September 11. Added clarifications in the text on the bias modelling and compression limits following referee's comments. Removed tetraspectrum term from the pk-bk cross covariance + correction in the appendi